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Series Analysis of Randomly Diluted Nonlinear Networks with Negative Nonlinearity Exponent University of Pennsylvania ScholarlyCommons Department of Physics Papers Department of Physics 9-1-1987 Series Analysis of Randomly Diluted Nonlinear Networks With Negative Nonlinearity Exponent Yigal Meir Raphael Blumenfeld A. Brooks Harris University of Pennsylvania, [email protected] Amnon Aharony Follow this and additional works at: https://repository.upenn.edu/physics_papers Part of the Physics Commons Recommended Citation Meir, Y., Blumenfeld, R., Harris, A., & Aharony, A. (1987). Series Analysis of Randomly Diluted Nonlinear Networks With Negative Nonlinearity Exponent. Physical Review B, 36 (7), 3950-3952. http://dx.doi.org/ 10.1103/PhysRevB.36.3950 At the time of publication, author A. Brooks Harris was affiliated withel T Aviv University, Tel Aviv, Israel. Currently, he is a faculty member in the Physics Department at the University of Pennsylvania. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/306 For more information, please contact [email protected]. Series Analysis of Randomly Diluted Nonlinear Networks With Negative Nonlinearity Exponent Abstract The behavior of randomly diluted networks of nonlinear resistors, for each of which the voltage-current relationship is |V|=r|I|α, where α is negative, is studied using low-concentration series expansions on d-dimensional hypercubic lattices. The average nonlinear resistance ⟨R⟩ between a pair of points on the same cluster, a distance r apart, scales as rζ(α)/ν, where ν is the correlation-length exponent for percolation, and we have estimated ζ(α) in the range −1≤α≤0 for 1≤d≤6. ζ(α) is discontinuous at α=0 but, for α<0, ζ(α) is shown to vary continuously from ζmax, which describes the scaling of the maximal self-avoiding-walk length (for α→0−), to ζBB, which describes the scaling of the backbone (at α=−1). As α becomes large and negative, the loops play a more important role, and our series results are less conclusive. Disciplines Physics Comments At the time of publication, author A. Brooks Harris was affiliated withel T Aviv University, Tel Aviv, Israel. Currently, he is a faculty member in the Physics Department at the University of Pennsylvania. This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/306 PHYSICAL REVIEW B VOLUME 36, NUMBER 7 1 SEPTEMBER 1987 Series analysis of randomly diluted nonlinear networks with negative nonlinearity exponent Yigal Meir, Raphael Blumenfeld, A. Brooks Harris, * and Amnon Aharony Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel (Received 4 May 1987) The behavior of randomly diluted networks of nonlinear resistors, for each of which the voltage-current relationship is V =r where is is studied low- I I I I I, a negative, using concentration series expansions on d-dimensional hypercubic lattices. The average nonlinear resis- tance &R) between a pair of points on the same cluster, a distance r apart, scales as r~ ' ", where v is the correlation-length exponent for percolation, and we have estimated g(a) in the range —1 ~ a ~ 0 for 1 ~ d ~ 6. ((a) is discontinuous at a =0 but, for a (0, g(a) is shown to vary continuously from g,„, which describes the scaling of the maximal self-avoiding-walk length (for a 0 —), to gas, which describes the scaling of the backbone (at a= —1). As a becomes large and negative, the loops play a more important role, and our series results are less conclusive. Kenkel and Straley' proposed a model of networks in N,„—L'. Based on available values for g(a)—:g(a)v, which bonds on percolation clusters consist of nonlinear where v is the exponent that describes the scaling of the resistors, each of which obeys the generalized Ohm's law correlation length, at a =1,0+, and —1, we constructed an approximant function for g(a), which is reproduced AV=r I I I 'sgnI, (for d=2) in Fig. l. where hV is the voltage drop across the resistor, I is In this work we carry on the series expansion described current Bowing through it, r is the nonlinear resistance, in Ref. 3 (later referred to as I) to negative a, in order to and a is the exponent characterizing the nonlinearity. In- obtain estimates of ((a) for all d and a. terest in this model centers on the region of concentration The percolation susceptibility is defined by p for p-p„where p, is the critical concentration for per- colation. vji (2) Blumenfeld and Aharony showed that the nonlinear av resistance R,(L) between two terminals on the same clus- where v;j is 1 if the two sites i and j belong to the same ter, a distance L apart, reduces to some geometrical cluster and zero otherwise, and [ l,„denotes an average characteristics of the cluster for specific values of a. For over all configurations of occupied and unoccupied bonds. the nonlinear resistance a 0+ describes the length of The nonlinear resistive susceptibility is defined by the minimal path between the two terminals, while for a ~ the resistance reduces to the number of singly con- Ztt(a) = gR;, (tt) v;, (3) nected bonds between the terminals. For a=1 the non- . J av linear resistance reduces trivially to the linear one. Conse- where R;J(a) is the nonlinear resistance between i and j. quently, the exponent g(tt), which describes how the non- In (3) we interpret to be zero when =0 L~' Eq. R;I(a) vj v;J linear resistance R,(L) — scales with the distance L and R;z (a) =~. for L«g, where g is the percolation correlation length, Section II of I describes in detail the construction of the should reduce in the above limits to the exponents that de- scribe the scaling of the corresponding geometrical quan- tities. These results were confirmed by series expansions and the e expansion. 3.0 Recently, we proposed that the above results could be extended to negative a, where again the resistance for 25 some particular values of a corresponds to other geometri- cal characteristics of the network. In particular, for a 0 — the resistance reduces to the maximal self- = — avoiding path between the terminals; for a 1 the resis- 15— tance describes the number of backbone bonds (the num- ber of bonds that carry current), while for a —~ the I,O resistance scales with an exponent z I a where z de- I, —2.0 IO 20 scribes the scaling of the maximal "cutting surface" of the backbone between the terminals, i.e., the largest number of bonds, N, „, which one can cut in order to break the FIG. 1. Approximant function for g(a) at d=2 (from Ref. 5) backbone into two pieces, each connected to one terminal, (solid line), and series results (solid circles). 36 3950 1987 The American Physical Society 36 BRIEF REPORTS 3951 series and we will not repeat it here. However, for nega- reduce to the series for the length of the maximal self- tive a there is an additional complexity. As described in avoiding walk between the two terminals on the cluster, Ref. 5, one may find a number of solutions to the non- which we constructed independently. In Table I, we give linear KirchhoA"s equations, each of which corresponds to the coefficients A(a, k, l) of the series for the nonlinear a given allowed assignment of directions of the currents on resistance susceptibility for a = —2, based on the solu- the bonds. Each solution corresponds to a local extremum tion which minimizes the power P. of the power In order to analyze the series, we divided the coefficients of each series (for any a), by those of the series for the percolation susceptibility, E~, term by term. One can a+1 b a+1 show (see, e.g., Ref. 8) that if two series diverge at the (4) same critical point, the series that results from dividing one by the other, term by term, diverges at p = 1 no matter where the sum is over occupied bonds b (No. te that this what the value of the critical point was, with an exponent definition is slightly diA'erent from that of the power in that is equal to the diA'erence between the two exponents Ref. 5). Allowed regions 0 in voltage space correspond to plus one. Since in our case we believe that the series diverge at p„and the difference between the exponents is choosing current directions such that h, V is irrotational = and has no internal sources or sinks. Each region 0 has a g(a), the constructed series is expected to diverge at p 1, boundary 8 on which one or more of the h, Vb's vanish, so with an exponent P(a) + l. In this way, we obtain an esti- that for —1 & a &0 P is infinite on 8. Thus, for this mate for ((a) which is not biased by the value of p, nor by range of a each region 0 has a local minimum of P corre- the value of y~, the exponent that describes the divergence sponding to a solution to Kirchhoff's equations, of X~. The resulting series were analyzed by the nonho- 8P/8V; 0. Since there is a priori no reason to choose mogeneous differential Pade method. The results of the any one of these solutions as the "physical" one, we con- analysis, however, are less—conclusive than those for posi- structed the series both for the solution that gives the tive a. In the range 1 ~ a~0 one can still obtain = — ——, minimal power and for the one that gives the maximal reasonable results, and those for a 1, ', 0, and power.
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